# Realizing Stiefel-Whitney classes via vector bundles

Let $$X$$ be a CW complex. If $$E$$ is a vector bundle over $$X$$, then it's well-known that the Stiefel-Whitney classes $$w_j(E) \in H^j(X,\mathbb F_2)$$ of $$E$$ are determined from the classes $$w_{2^k}(E)$$ (for $$2^k \leq j$$) via the Wu formula, using the cup product and the action of the Steenrod algebra.

Question 1: Does the Wu formula imply any further relations? This is a purely algebraic question which I make more precise in (a) and (b) below.

That is, let $$H$$ be a nonnegatively-graded $$\mathbb F_2$$-algebra with an unstable action of the Steenrod algebra satisfying the Cartan formula and $$Sq^{|x|}(x) = x^2$$ for all homogenenous $$x \in H$$. Let $$W$$ be the set of sequences $$(w_j \in H^j)_{j \in \mathbb N}$$ with $$w_0 = 1$$ and satisfying the Wu formla.

(a) For any sequence $$(v_{2^k} \in H^{2^k})_{k \in \mathbb N}$$, does there exist $$w \in W$$ (necessarily unique) with $$w_{2^k} = v_{2^k}$$ for all $$k \in \mathbb N$$?

Presumably (i) the Whitney sum formula and (ii) the universal formula for the Stiefel-Whitney classes of a tensor product of vector bundles are compatible with the Wu formula, so that $$W$$ is a commutative ring using (i) for addition and (ii) for multiplication.

(b) Is $$W$$ a polynomial algebra on whichever generators from (a) do exist?

Question 2: What restrictions beyond the Wu formula are there restricting the Stiefel-Whitney classes of a vector bundle $$E$$ on a CW complex $$X$$? This is a genuinely topological question.

Over here Mark Grant describes one such restriction, but ideally I'd like a more systematic discussion.

If it simplifies matters to assume that $$X$$ is finite, or even a compact manifold, then that's fine.

• I once asked a related question: mathoverflow.net/questions/239482/… – Jens Reinhold Jul 3 '20 at 6:53
• Isn't Question 2 answered by the integral cohomology of $BO(n)$? – John Greenwood Jul 3 '20 at 15:55
• @JohnGreenwood I'm not sure what you mean? I agree that the integral cohomology of $BO(n)$ will pull back to "integral characteristic classes" which probably contain some information that the Stiefel-Whitney classes don't. Do you have something more specific in in mind? – Tim Campion Jul 3 '20 at 18:34